Optimal Dynamic Advertising Policies in Digital and Traditional Channels: A Control-Theoretic Approach.

This article investigates the optimal allocation of advertising efforts between digital and traditional channels for a monopolistic firm using optimal control theory. It explicitly models the substitution effect—where advertising in one channel reduces the marginal return of advertising in another—and incorporates different decay rates of goodwill generated by each channel through an integro-differential equation framework. The analysis reveals that digital advertising, due to its lower decay rate, always warrants investment throughout the planning horizon, while traditional advertising should be employed primarily when digital’s comparative advantage diminishes over time. Additionally, when synergistic effects between channels are considered, firms tend to adopt traditional advertising earlier and may maintain positive investment in both channels simultaneously, with the allocation ratio dynamically shifting depending on the intensity of synergy. Extensions to three-channel models confirm these findings, emphasizing the importance of dynamically adjusting advertising strategies over time to maximize profits.